docs: def pdf_Gates_Gaudin_Schuhman(d, d_characteristic, m):
      r'''Calculates the probability density of a particle
      distribution following the Gates, Gaudin and Schuhman (GGS) model given a 
      particle diameter `d`, characteristic (maximum) particle
      diameter `d_characteristic`, and exponent `m`.
      
      .. math::
          q(d) = \frac{n}{d}\left(\frac{d}{d_{characteristic}}\right)^m 
          \text{ if } d < d_{characteristic} \text{ else } 0
          
      Parameters
      ----------
      d : float
          Specified particle diameter, [m]
      d_characteristic : float
          Characteristic particle diameter; in this model, it is the largest 
          particle size diameter in the distribution, [m]
      m : float
          Particle size distribution exponent, [-]    
  
      Returns
      -------
      pdf : float
          GGS probability density function, [-]
  
      Notes
      -----
      The characteristic diameter can be in terns of number density (denoted 
      :math:`q_0(d)`), length density (:math:`q_1(d)`), surface area density
      (:math:`q_2(d)`), or volume density (:math:`q_3(d)`). Volume density is
      most often used. Interconversions among the distributions is possible but
      tricky.
  
      Examples
      --------
      >>> pdf_Gates_Gaudin_Schuhman(d=2E-4, d_characteristic=1E-3, m=2.3)
      283.8355768512045
  
      References
      ----------
      .. [1] Schuhmann, R., 1940. Principles of Comminution, I-Size Distribution 
         and Surface Calculations. American Institute of Mining, Metallurgical 
         and Petroleum Engineers Technical Publication 1189. Mining Technology, 
         volume 4, p. 1-11.
      .. [2] Bayat, Hossein, Mostafa Rastgo, Moharram Mansouri Zadeh, and Harry 
         Vereecken. "Particle Size Distribution Models, Their Characteristics and
         Fitting Capability." Journal of Hydrology 529 (October 1, 2015): 872-89.
      '''
      if d <= d_characteristic:
          return m/d*(d/d_characteristic)**m
      else:
          return 0.0